J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 355 



mean of the squares of the errors ; and, generally, e„ Is the wth 

 root of the arithmetical mean of the ^th powers, without regard 

 to signs. 



We see from the numerical part of the hmiting values, that 

 the determination by the sum of the squares is the most advan- 

 tageous one. With an equal number of observations, we ob- 

 tain by its means the narrowest limits within which there arc 

 equal chances that r lies. The number of observations neces- 

 sary for. attaining equal limits, according as we employ s^, e^ £3, 

 &c., will be to each other as 



_ o . 1 . 15?r-8 . £ . 945 tt— 128 _ 113 _ 

 " " ~36 ■ 3 * 1600 * 45 ' 

 or if with ?^ one hundred observations are required to attain 

 certain hmits, there are I'equired for the same limits, with 

 Sj 114 observations 



S3 . • 109 



e 133 „ 



«6 



178 

 251 



On account of the great convenience of Sj, and the not very con- 

 siderable difference in the naiTowness of the limits, the employ- 

 ment of Sj will be most frequently preferred if the sum of the 

 squares of the eiTors is not already known. 



For the above example the sum of the absolute eiTors 

 = 181"898 ; consequently, 



,= 15^ = 6-496, 



and thence 



r = 5"'-492 

 within the limits 



4"'-972 and 6"'-012, 



a value which, if it differs from that above given, still leads, for 

 the small number of observations, to a sufficient estimation of 

 the accuracy of the result. 



We may employ besides for this determination the proposi- 

 tion which has no direct reference to the magnitude of the 

 single errors, but only declares that, according to the universal 

 law of probability [without determinate assumption of <|> (A)], 

 the idea of the probaljle error contains the condition that 

 there occur as many errors less than r, as there are greater. 

 If, therefore, we arrange the errors, without regard to their 



